Abstract
The problem of “quantum ergodicity” addresses the limiting distribution of eigenfunctions of classically chaotic systems. I survey recent progress on this question in the case of quantum maps of the torus. This example leads to analogues of traditional problems in number theory, such as the classical conjecture of Gauss and Artin that any (reasonable) integer is a primitive root for infinitely many primes, and to variants of the notion of Hecke operators.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Bouzouina and S. De BièvreEquipartition of the eigenfunctions of quantized ergodic maps on the torusComm. Math. Phys. 178 (1996), 83–105.
Y. Colin de VerdièreErgodicité et fonctions propres du laplacienComm Math. Phys. 102 (1985), 497–502.
M. Degli EspostiQuantization of the orientation preserving automorphisms of the torusAnn. Inst. Poincaré 58 (1993), 323–341.
M. Degli Esposti, S. Graffi and S. IsolaClassical limit of the quantized hyperbolic toral automorphismsComm. Math. Phys. 167 (1995), 471–507.
J. H. Hannay and M. V. BerryQuantization of linear maps on a torus - Fresnel diffraction by a periodic gratingPhysica D 1 (1980), 267–291.
J. Keating and F. MezzadriPseudo-Symmetries of Anosov Maps and Spectral StatisticsNonlinearity 13 (2000), no. 3, 747–775.
S. KnabeOn the quantisation of Arnold’s catJ. Phys. A: Math. Gen. 23 (1990), 2013–2025.
P. Kurlberg and Z. RudnickHecke theory and equidistribution for the quantization of linear maps of the torusDuke Math.J. 103 (2000), no. 1, 47–77.
P. Kurlberg and Z. RudnickOn quantum ergodicity for linear maps of the toruspreprint math/9910145, to appear in Comm Math. Phys.
J. Marklof and Z. RudnickQuantum unique ergodicity for parabolic mapsGeom. and Funct. Analysis 10 (2000), no. 6, 1554–1578.
M. MurtyArtin’s conjecture for primitive rootsMath. Intelligencer 10 (1988), no. 4, 59–67.
Z. Rudnick and P. SarnakThe behaviour of eigenstates of arithmetic hyperbolic manifoldsComm. Math. Phys. 161 (1994), 195–213.
A. SchnirelmanErgodic properties of eigenfunctionsUsp. Math. Nauk 29 (1974), 181–182.
S. ZelditchUniform distribution of eigenfunctions on compact hyperbolic surfacesDuke Math. J. 55 (1987), 919–941.
S. ZelditchQuantum ergodicity of C* -dynamical systemsComm. Math. Phys. 177 (1996), 507–528.
S. ZelditchIndex and dynamics of quantized contact transformationsAnn. Inst. Fourier (Grenoble) 47 (1997), 305–363.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Rudnick, Z. (2001). On Quantum Unique Ergodicity for Linear Maps of the Torus. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_37
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8266-8_37
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9496-8
Online ISBN: 978-3-0348-8266-8
eBook Packages: Springer Book Archive